Is the effect part of the guitarist's sound? Guitarists spend tons of money and time getting very specific sounds. I mean, there are tons of websites, forums, magazines, manufacturers, trade shows, and urban myths built around guitarists and their effects. Their sound might be of a TC Electronics SCF pedal, and you're not going to get that sound with a plugin after the fact. Also, the order of effects is of tremendous importance! That includes the amp! If you run an effect first and then into an amp and record it, it's gonna sound WAY different than recording the amp and then applying the effect later. Plus there's the whole digital versus analog thing. How many people think hardware compressors sound way better than plugins? Same with effects.
I'm not saying you can't record dry and add effects later, but I wouldn't call it what people usually do. They're two different techniques with two very different results, and there's a time and place for each, but one is a poor substitute for the other.
As far as what pedals you can't live without, that's going to depend wholly on the guitarist you're talking about. I need my Big Muff Pi, Tubescreamer, and Line 6 DL-4. I've got other pedals, but I'd never consider trying to replace them with effects placed after tracking (although I am going to replace the DL-4 with the new TC Electronics X4 when it comes out.) Other guitarists with other styles will have different tastes and thus different answers. It's a bit like asking someone what's the best color, if you could have only one.
A lot of guitarists these days are all about boutique pedals. I can certainly understand why. A big deal in the guitar world is establishing a unique sound, so that anyone can recognize your "voice". That's a large reason why just about every effect I own has been modded to some extent. Also, I'm less concerned about owning versatile pedals, and more concerned about owning pedals that precisely nail that sound in my head. I rarely touch any dials on the effect once I've dialed in my favorite sound.
Now, having gone off on that spiel, I should at least try to be more helpful and less egomaniacal. I wouldn't look to specific brands or pedal names to start off with, and instead look to categories of effects. There are tons of different types of effects, and it would save a lot of time and money if you could narrow down which particular style of effects you would consistently use. From there, you could narrow down the exact brand and model of effect. Here's a list of my most used effects:
Fuzz pedal (different from overdrive)
Chorus pedal (or a flanger. they're not identical, but cover some similar ground)
I left quite a few out like octave pedals, phaser pedals, synths, etc... But one of each of the above would be what I would call a minimum pedal board.